I presented this poster, “Static vs Falling: Time slicings of Schwarzschild black holes” at the GR21 conference in New York City, July 2016. You can download a PDF version. A big thank-you to those who gave feedback, especially to Jiří Podolský who encouraged me to publish it! The section on coordinate vectors has been updated. Also there is a major update to the poster.

The following is a natural choice of orthonormal tetrad for an observer moving radially in Schwarzschild spacetime with “energy per unit rest mass” e:

The components are given in Schwarzschild coordinates. (The ± signs are not independent — they must be either both +1 or both -1. Note that e does not distinguish between inward and outward motion. There is additional freedom to define any of these vectors as their negative.)

We normally think of e as invariant, where there is a presumption of freely falling / geodesic motion, but even if not we can regard it as an instantaneous value.

is the 4-velocity computed previously. The other vectors can be obtained from substituting and into the tetrad here. is determined from and the equation for e above, then V follows from inverting . This orthonormal frame is useful for determining the object’s perspective, e.g. tidal forces, visual appearances, etc.

Suppose an observer u moves radially with speed (3-velocity) relative to “stationary” Schwarzschild observers, where we define as inward motion. Then one natural choice of orthonormal tetrad is:

where the components are given in Schwarzschild coordinates. This may be derived as follows.

The Schwarzschild observer has 4-velocity

because the spatial coordinates are fixed, and the t-component follows from normalisation (Hartle §9.2).

Now the Lorentz factor for the relative speed satisfies , and together with normalisation and the assumption that the θ and φ components are zero, this yields given above.

We obtain by orthonormality: and , and again making the assumption the θ and φ components are zero. Note the negative of the r-component is probably an equally natural choice. Then and follow from simply normalising the coordinate vectors.

Strictly speaking this setup only applies for , because stationary timelike observers cannot exist inside a black hole event horizon! Yet remarkably the formulae can work out anyway (MTW …) . An alternate approach is local Lorentz boost described shortly.

A nice way to parametrise the 4-velocity u of a small test body moving radially within Schwarzschild spacetime is by the “energy per unit rest mass” e:

For the “±” term, choose the sign based on whether the motion is inwards or outwards. All components are given in Schwarzschild coordinates . The result was derived as follows. In geometric units, the metric is:

By definition , where is the Killing vector corresponding to the independence of the metric from t, and has components (Hartle §9.3). For geodesic (freefalling) motion e is invariant, however even for accelerated motion e is well-defined instantaneously and makes a useful parametrisation.

We want to find say. Rearranging the defining equation for e gives . Radial motion means , so the normalised condition yields the remaining component . The resulting formula is valid for all , and for e=1 the 4-velocity describes “raindrops” as expected.